Question

If α,β,γ are the roots of the equation 5x3─8x2 + 7x + 6 = 0
Find the equation whose roots are
α2 + β2 + βα , β2 + γ2 + γβ, γ2 + α2 + γα

Answer 1

 Here's one way to do it, but not the quickest way: 

5x^3 - 8x^2 + 7x + 6 = 0 is the same as x^3 - (8/5)x^2 + (7/5)x + (6/5) = 0. If α,β,γ are the roots, then this should be the same as (x-α)(x-β)(x-γ)=0. You can multiply this out and match coefficients, then find the variables. 

Similarly, an equation with roots α^2 + β^2 + αβ, β^2+γ^2+βγ, and γ^2 is 
(x - (α^2 + β^2 + αβ)) (x - (β^2+γ^2+βγ)) (x-γ^2) = 0 
(x - (α+β)^2 + αβ)) (x - (β+γ)^2 - βγ)) (x-γ^2) = 0